Schreiber
master thesis Bongers

A thesis that I once supervised:


on the modern formulations of geometric quantization and leading towards higher geometric quantization.


Contents

Abstract

The first part of this thesis provides an introduction to recent developments in geometric quantization of symplectic and Poisson manifolds, including modern refinements involving Lie groupoid theory and index theory/K-theory. We start by giving a detailed treatment of traditional geometric quantization of symplectic manifolds, where we cover both the quantization scheme via polarization and via push-forward in K-theory. A different approach is needed for more general Poisson manifolds, which we treat by the geometric quantization of Poisson manifolds via the geometric quantization of their associated symplectic groupoids, due to Weinstein, Xu, Hawkins, et al.

In the second part of the thesis we show that this geometric quantization via symplectic groupoids can naturally be understood as an instance of higher geometric quantization in higher geometry, namely as the boundary theory of the 2d Poisson sigma-model. This thesis closes with an outlook on the implications of this change of perspective.

The following provides some hyperlinked keywords as to the content of the thesis.

1. Introduction

2. Geometric quantization of Symplectic Manifolds

3. Geometric quantization of Poisson Manifolds

4. Higher geometric perspective

4.1 Motivating examples

(Fiorenza-Sati-Schreiber 12, Fiorenza-Sati-Schreiber 13)

4.2 Higher prequantum geometry

(Fiorenza-Rogers-Schreiber 13)

4.3 Higher symplectic geometry

4.3.1 Simplicial de Rham cohomology

4.3.2 Lie \infty-Algebroids

4.3.3 Cocycles, invariant polynomials and Chern-Simons elements

4.3.4 Poisson σ\sigma-model

4.3.5 Lie \infty-integration

4.3.6 2d Poisson-Chern-Simons theory

4.3.7 Boundary field theory

This perspective suggests a more encompassing perspective in which geometric quantization of symplectic groupoids is a realization of the holographic principle in local quantum field theory and a non-perturbative refinement of how deformation quantization of Poisson manifolds is the perturbative boundary field theory of the perturbative Poisson sigma-model.

Expanding on this is the topic of the Outlook section.

5. Outlook

References

Last revised on April 1, 2017 at 11:36:39. See the history of this page for a list of all contributions to it.